To find the ground speed and actual bearing of the plane, we can use the concept of vector addition.
1. Start by understanding the given information:
- Airspeed: 192 mi/h (speed of the plane with respect to the air)
- Bearing: 121 degrees (direction in which the plane is heading)
- North wind: 15.9 mi/h (speed of the wind blowing from north to south)
2. Break down the airspeed vector into its components:
- Horizontal component: Airspeed * cos(Bearing)
Ground speed in the eastward direction = 192 mi/h * cos(121°)
- Vertical component: Airspeed * sin(Bearing)
Ground speed in the northward direction = 192 mi/h * sin(121°)
3. Add the wind vector to the ground speed components:
- Eastward ground speed = Ground speed in the eastward direction + Wind speed in the eastward direction
- Northward ground speed = Ground speed in the northward direction + Wind speed in the northward direction
4. Calculate the magnitudes of the ground speed:
- Ground speed = Square root of (Eastward ground speed^2 + Northward ground speed^2)
5. Calculate the actual bearing of the plane:
- Actual bearing = arctan(Northward ground speed / Eastward ground speed)
Now, let's calculate the values:
Horizontal component:
Ground speed in the eastward direction = 192 mi/h * cos(121°)
Ground speed in the eastward direction ≈ 192 mi/h * (-0.4848)
Ground speed in the eastward direction ≈ -93.2 mi/h
Vertical component:
Ground speed in the northward direction = 192 mi/h * sin(121°)
Ground speed in the northward direction ≈ 192 mi/h * (0.8746)
Ground speed in the northward direction ≈ 167.8 mi/h
Adding wind vector to ground speed components:
Eastward ground speed = -93.2 mi/h + 0 mi/h (since wind blows north-south)
Northward ground speed = 167.8 mi/h - 15.9 mi/h (subtracting wind speed from northward ground speed)
Calculating the magnitude of the ground speed:
Ground speed = Square root((-93.2 mi/h)^2 + (167.8 mi/h - 15.9 mi/h)^2)
Ground speed ≈ Square root(8657.24 + 2645.44)
Ground speed ≈ Square root(11302.68)
Ground speed ≈ 106.37 mi/h
Calculating the actual bearing of the plane:
Actual bearing ≈ arctan((167.8 mi/h - 15.9 mi/h) / -93.2 mi/h)
Actual bearing ≈ arctan(151.9 mi/h / -93.2 mi/h)
Actual bearing ≈ arctan(-1.63)
Since the tangent function is negative, the actual bearing is in the third or fourth quadrants.
Actual bearing ≈ 189.11°
Therefore, the ground speed of the plane is approximately 106.37 mi/h, and the actual bearing is approximately 189.11 degrees.